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Show that a map $F$ between smooth manifolds $M\subset \Bbb R^n$ and $N\subset \Bbb R^k$ is smooth iff $f\circ F$ is a smooth function on $M$ whenever $f$ is smooth on $N$.

My Attempt:
I'm having trouble in the "$\impliedby$" direction, namely, how to show that ($f\circ F$ is a smooth function on $M$ whenever $f$ is smooth on $N$)$\implies$ ($F$ smooth).

Now given $x\in M$, it suffices to show there exist charts $\phi:U\to V,\,\eta:W\to Z$ such that $x\in U,\, F(x)\in W$, such that \begin{equation}\eta\circ F\circ \phi^{-1}:\phi(U\cap F^{-1}(W))\to Z\tag{1}\end{equation} is smooth. (Existence suffices; or you might just take it as the definition of smooth maps between manifolds.)

The assumed condition here is $f\circ F$ is smooth if $f$ is so. But $f\circ F$ being smooth just means that $f\circ F\circ \phi^{-1}$ is smooth. In light of $(1)$, it is naturally that I want to somehow connect $\eta$ with a smooth function $f$ on $N$.

My initial thoughts were to break down $\eta$ into its components, each of which should be "smooth". The gap here is that $\eta$ is a chart which is only a locally defined thing but $f$ by our requirement must be a globally defined function on $N$. Indeed, it is easy for me to find functions $f_i:=\pi_i\circ\eta_i$ ($\pi_i$ is the coordinate function in $\Bbb R^k$, and $\eta$ is a chart at $F(x)$), which is by definition locally smooth at $F(x)$. So we have $(f_1,\cdots,f_k)^T = \eta$ smooth, and hence $\eta\circ F\circ\phi^{-1}$ is by definition locally smooth at $\phi(x)$ and we're done. However, it's just not obvious how to extend these locally defined $f_i$ to the whole $N$. I even doubt if it's in general possible.

So is there any alternative method to prove this direction? Thanks in advance.

1 Answers 1

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Vim, I think your method works fine. To make your "coordinate function" $f_i$ smooth globally, I suggest multiplying by a "bump function" localised around $F(x)$, i.e. a function with support contained inside your chosen chart around $F(x)$ and identically equal to one in a little closed neighbourhood around $F(x)$. Such bump functions are guaranteed to exist. This will give you something that looks like your coodinate function in the little closed neighbourhood of $F(x)$ but can be smoothly extended by zero away from your chart.

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    Could you be more specific about your bump function? First i think they should be defined on the image neighbourhood of $F(x)$ in $\Bbb R^k$ where things are good rather than directly around $F(x)$; second, i tried a mollifer $\mu$ with compact support around $\eta(F(x))$, and ended up trying to show $\mu\eta$ (product, not composition) is a local homéomorphism at $F(x)$, is this what you mean? – 2017-02-25
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    Yes, the bump function is defined on $\mathbb R^k$. A mollifer shaped like a bell curve won't do. You want something perfectly flat and equal to one in a neighbourhood of $\eta(F(x))$. – 2017-02-25